Risks 2015, 3, 318-337; doi:10.3390/risks3030318OPEN ACCESS

risksISSN 2227-9091

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Delivering Left-Skewed Portfolio Payoff Distributions in thePresence of Transaction Costs †

Jacek B Krawczyk

Victoria University of Wellington, School of Economics and Finance, PO Box 600, Wellington 6140,New Zealand; E-Mail: J.Krawczyk@vuw.ac.nz; Tel.: +644-463-5352.

† This paper is an extended version of a paper under the same title presented by the author at theQuantitative Methods in Finance 2013 Conference, Sydney, Australia. It was further revised while theauthor was a visiting scholar at the Department of Mathematics at Macquarie University, Sydney,NSW 2109.

Academic Editor: Andrea Consiglio

Received: 10 January 2015 / Accepted: 11 August 2015 / Published: 21 August 2015

Abstract: For pension-savers, a low payoff is a financial disaster. Such investors willmost likely prefer left-skewed payoff distributions over right-skewed payoff distributions.We explore how such distributions can be delivered. Cautious-relaxed utility measures arecautious in ensuring that payoffs don’t fall much below a reference value, but relaxed aboutexceeding it. We find that the payoff distribution delivered by a cautious-relaxed utilitymeasure has appealing features which payoff distributions delivered by traditional utilityfunctions don’t. In particular, cautious-relaxed distributions can have the mass concentratedon the left, hence be left-skewed. However, cautious-relaxed strategies prescribe frequentportfolio adjustments which may be expensive if transaction costs are charged. In contrast,more traditional strategies can be time-invariant. Thus we investigate the impact oftransaction costs on the appeal of cautious-relaxed strategies. We find that relatively hightransaction fees are required for the cautious-relaxed strategy to lose its appeal. This papercontributes to the literature which compares utility measures by the payoff distributions theyproduce and finds that a cautious-relaxed utility measure will deliver payoffs that manyinvestors will prefer.

Keywords: portfolio management; payoff distributions; pension funds; transaction costs

MSC classifications: 93E20, 93E25, 90C39, 90C40, 90A09

JEL classifications: G11, C61, J32

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1. Introduction

This paper deals with problems concerning loss-avoiding investors. For these investors, the highprobability of a loss, manifested by a positively skewed payoff distribution1, is unacceptable even with achance of big gains.

A preference against skewness or, in other words, a preference for negative (left) skewness may goagainst the norm in the literature. This is because the literature deals predominantly with gamblerswho are investors that are drawn to the lottery-like characteristic of securities that in odd cases canprovide extremely high returns. In particular, [1] and [2] find that cumulative prospect theory can explaininvestor’s desire for positive skewness, which can lead to right skewed securities being over-priced. Theinterest of this paper is in anti-gamblers who are investors also attracted to uncertain payoff securitiesbut such that they do not seek extremely high returns.

An example of an anti-gambler is a pension saver. Pensioners may be entirely reliant on their savingsto support themselves, and thus a small payoff could be disastrous. The authors of [3] discovered caseswhere risk averse investors prefer negative skewness. These investors cannot be represented by eitherthe Merton utility function (see [4]) or the functions from prospect theory (for typical parameter values).

It is the interest in the distribution skewness that differentiates this paper (also, that of [5] and severalothers cited in footnote 6) from other research in the area of dynamic portfolio management.2 Infact, the literature tends to ignore the distribution of actual outcomes that eventuate from the optimalinvestment strategy.3

Traditionally, an investor’s utility function is based on psychological experiments and perceivedpreferences. This is then used to derive a functional form for the utility function and a portfoliooptimisation problem is solved based on this measure. The obtained strategy is optimal for the adoptedutility function. The authors of [6] take the opposite approach: associate a utility function with a givenpayoff distribution. So, they can design a utility function to be optimised, for an educated anti-gamblerwho could specify their desired investment-outcome spread. Studying the distributions of outcomesthat arise from following various strategies helps compare those strategies and the utility measuresthat generate them. Then, one can come to a conclusion about the appropriateness of each utilitymeasure. This conclusion agrees with [7]’s statement that [a]vailable empirical evidence suggests thatthe distribution of losses plays an important role in understanding the preferences of individuals because,evidently, the distributions of outcomes tell us about possible losses of investors.

1 (1) When the mass of the distribution is concentrated on the left and the right tail is longer, the distribution is said right- orpositively skewed; (2) when the mass of the distribution is concentrated on the right the left tail is longer, the distributionis said left- or negatively skewed.

2 Interestingly, in the context of static portfolio management, [8] also question the wisdom of “traditional optimisation" forsome investors, leverage-averse in their case.

3 Exceptions to this include [9], [10]. The work done on the distribution builder in the first, enables subjects to build theirdesired pension distribution subject to a budget constraint. However, the results lead to distributions that are right skewed,which could be due to the setting of a (low) reference point at the amount guaranteed by the risk-free asset. In [10],quantiles are proposed as an effective way to evaluate the success and failings of a portfolio.

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The focus on the distribution of payoffs in dynamic portfolio management can be compared to thaton the efficient frontier in standard mean-variance optimisation. In a static model, almost everythingand anything can be explained by the efficient frontier analysis. A cautious investor will immediatelyrecognise which strategy they like, from analysing the frontier points obtained for different investments.This cannot be done for a dynamic model considered in this paper because mean-variance optimisationinvolves the variance term whose computation cannot be separated from mean maximisation, in the senseof dynamic programming. Providing a dynamic portfolio investor with a payoff distribution overcomesthis difficulty.

The utility measures proposed in this paper are non-symmetrical with respect to risk,non-differentiable at a reference payoff and only locally concave. This hinders us in obtaining closedform solutions4. Thus, our solutions will be numerical and parameter-specific. We will study ahypothetical base-case scenario5 that involves an initial outlay invested in a pension fund to grow and becollected as a lump sum after a given optimisation horizon.

Following [11]6, it is proposed in this paper that by optimising a utility measure, which has areference-payoff level and which is concave on each side to this level, new “cautious-relaxed" strategiesmay be obtained. A cautious-relaxed strategy will be adopted by agents who are cautious in ensuringthat payoffs do not fall much below a reference value, but relaxed about exceeding that reference value.In brief, it will generate a left-skewed payoff distribution, as opposed to the right-skewed ones that resultfrom maximisation of a usual risk-averse (concave) utility function. A cautious-relaxed strategy satisfiesthe needs of loss-avoiding investors.

As it will be shown in this paper, such a strategy consists of buying more shares when the fundstarts to perform poorly and investing into a secure asset when the fund performs well. That mayrequire frequent rebalancing of the portfolio, which can be costly. Intuitively, with high transactioncosts, a cautious-relaxed strategy will lose its appeal because the costs will nullify the gains related toleft-skewness of the payoff distribution. To confirm this intuition, in this paper (unlike in [11] and itspredecessors) portfolio rebalancing costs will be included. The profitability of cautious-relaxed strategieswill thus be assessed in a more realistic environment than before.

The paper proceeds as follows. Sections 2, 3 and 4 discuss the problem of a pension fundoptimisation. They deal with wealth dynamics, performance measures and investment strategiesrealisations respectively. In Section 5 a sensitivity analysis of the pension yield to transaction costsis performed. This enables us to conclude in Section 6 that the desired features of cautious-relaxedstrategies are preserved for a level of costs that can mitigate the control jerkiness caused byportfolio rebalancing.

4 Finding analytic solutions to the resulting PDEs would be a substantive research project which may not yield any resultsas the study subject are nonlinear PDEs. A “semi" analytic solution could be obtained by a functional expansion. Wepursue numerical solutions in this paper, which are reliable and easy to interpret for parameter-specific problems.

5 Notwithstanding the obtained solution’s parameter-specificity, our analysis can be extended to other cases through the useof specialised software (see [12]).

6 Also [13], [14], [15] and [16] .

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2. Wealth dynamics

A plausible situation in financial management is one in which an investor deposits an amount x0

with a pension fund at time 0, to be repaid at time T . The investor’s wealth at time t is x(t) and sotheir pension is x(T ), a lump sum. Its amount depends upon both the investment strategy µ(x(t), t)

(t ∈ [0, T ]) adopted by the investor and market conditions. The market conditions are deterministicallyunpredictable so are usually modelled with stochastic processes. Consequently, the pension is a randomvariable and the pension fund problem is inherently stochastic.

Interdependence between an initial deposit x0, the pension collection time T , some reference payoffxT and the possible objective differences between the fund manager and the client in the context ofcautious-relaxed strategies have been discussed in [14] and [17]. Here, we assume that x0, T, xT aregiven and do not distinguish between the client and manager.

As is commonly the case in the literature (see [4], with further details available on pp. 160–161

of [18]), the portfolio consists of two assets, one risky7 (shares) and the other risk free (cash). Let theprice p(t) per share of the risky asset change according to the equation

dp(t) = αp(t)dt+ σp(t)dw (1)

where α, σ > 0 are constants and w is a one-dimensional standard Brownian motion. Let the priceq(t) per share of the risk-free asset change according to the equation

dq(t) = rq(t)dt,

where r ∈ (0, α) is a constant. Thus the fund value x(t) at time t ∈ [0, T ] changes according to thestochastic differential equation

dx(t) = (1− u(t))rx(t)dt+ u(t)x(t)(αdt+ σdw)− v(t)dt. (2)

Here, u(t) and 1−u(t) respectively denote the fractions of the fund invested in the risky and risk-freeassets at time t, and v(t) denotes the fund consumption rate. In this model, represented by equation (2),u(t) is the control variable. We will also assume that

0 6 u(t) 6 1 (3)

holds for all t ∈ [0, T ].8

In the real world fund investors incur some management fee which usually is a fixed proportion ofx(t). A possible reason for this may be that rather than use fee rates to maximise revenue the fundmanagement sets fees constant and uses information on past returns to attract investors. Hence the fundmanagers increase their revenue by increasing their client base, rather than increasing their fees. In otherwords even though all revenue may come from fees (so there is an incentive to change them) it is farmore important to have a large investor base which comes from having a good investment strategy.9 Inthis paper, we suppose a fee with force cx(t) will be charged, where c > 0 is a constant.

7 This will be a synthetic aggregate good if there are many risky assets.8 Constraint (3) means no short selling or borrowing. This restriction has been weakened in the literature; however, it may

be reasonable to keep it in a situation of a pension fund investor.9 A study of the impact of management incentives on investment strategies performed in [17] reports that maximising

management revenue from fees changes little the investment strategies.

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We will assume that the investor will also incur a transaction cost. The more shares are traded at onetime, the higher the cost. In this model, every portfolio adjustment will incur a cost proportional to theshare exposure’s correction |u(t)− u(t+ ∆)| (∆ > 0). Denoting the pace of correction by z(t), we get

z(t) ≡ εdu(t)

dt. (4)

The notation ‘ε× derivative’, where ε is a small and positive parameter, is borrowed from singularperturbation theory, see [19], and indicates that variable u(t) is “fast" and may not be differentiable inthe classic sense.

The investor will now control his wealth by selecting z(t), given the current level of shares u(t).10 Ifthe investor does not draw from the fund before T and the above fees are charged, wealth will evolveaccording to the following equation

dx(t) = (1− u(t))(r − c)x(t)dt− bx(t)|z(t)|dt+ u(t)x(t)((α− c)dt+ σdw) (5)

where b is a transaction price, r − c is the efficient cash (bond) rate and α− c is the risky asset drift. Noconstraint is imposed on the wealth x(T ) at time T , and – as said – the amount x0 of the initial depositand the fund management horizon T are taken as given.

3. Performance Measures

3.1. Aggregate Reward

The investment strategy governing a fund is, obviously, a function of the fund investor’s objectivefunction (or “performance measure"). Possible objectives include maximisation of the expected fundvalue, maximisation of the probability of achieving a target payoff and minimisation of shortfall, seee.g., [20]11. Once an objective function is proposed, the investor’s strategy can be computed as a solutionto a stochastic optimal control problem determined by the objective, where the system’s dynamics aregiven by (5), (4) and (3). The solution provides an optimal investment strategy µ(x(t), t) that generatescontrol u(t) = µ(x(t), t), also referred to as a strategy realisation. As argued in the introduction, thesolution should also include practical information about the resulting payoff distribution. This will allowthe investor to decide what they can reasonably expect for their pension, and find the objective functionthat satisfactorily represents their preferences. In particular, knowledge of the distribution of x(T ) isuseful to the investor, as it helps describe the risks associated with obtaining a particular realisation ofthe objective.

In general (see e.g., [18]), a pension fund investor that does not draw from the fund before T will finda strategy µ(x, t) maximising the total expected utility

J(x0, u0;µ) = E[h(x(T ), xT )

x(0) = x0, u(0) = u0

], (6)

10 An argument for using z(t) as control instead of u(t) can be found in [21]. If u(·) – the “fast" control – is Lebesguemeasurable, Proposition 3.2 in that publication establishes that a solution x(t) to a differential equation which containsu(t), but not z(t), can be approximated by a solution obtained from an equation where z(t) is introduced as in (4).

11 This publication deals with portfolio choice models for both pension funds and life assurance companies on a macro scalei.e., where many investors contribute to the fund. In that sense, our one-pension management problem is micro.

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where we have dropped the discounting term e−ρT (ρ > 0) as it is a constant number without an impact onthe strategy and payoff distribution. The stochastic optimal control problem of maximising (6) subject to(4)-(5)12 will be used in this paper as a model of the pension fund problem. Furthermore, we will assumethat the fraction of the fund invested in the risky asset at t = 0 is zero and drop u0 from the notation.

Now, we have to decide on the functional form of h(x(T )). It was shown in [13] that

hC(x(T ), xT ) :=

(x(T )− xT )κ if x(T ) > xT > 0, and

−(xT − x(T ))a otherwise,(7)

where a > 1 and κ ∈ (0, 1), captures the cautious-relaxed investor’s preferences for small lossesand high probability of a yield close to xT . As mentioned in the introduction, this utility measure isnon-differentiable at xT , concave on each side of this parameter and significantly non-symmetrical withrespect to risk.13

The payoff distribution corresponding to this utility measure will be contrasted in Section 4 with theMerton investor’s whose utility function is

hM(x(T )) =1

δ[x(T )]δ , 0 < δ < 1 , (8)

see [18]. This is the classic concave risk-averse utility function.The investor will maximise their utility measure

J(x0;µ) = E[h(x(T ))

x(0) = x0

]. (9)

subject to (4) - (5) (and (3)). A cautious-relaxed investor will use hC(·, ·) from (7); the Merton investorwill select hM(·) as (8).

3.2. Discussion

We will now comment on some properties of the final payoff utility measure (7), which is a kinked,two-piece power function, concave both above and below the kink (“double" concave), making it riskaverse everywhere.

We notice the majority of portfolios are optimised with respect to a smooth concave risk-averse utilityfunction à la [4] and that some formulations include constraints (see e.g., [23] and [24]). Using aprospect-theory utility function, which is convex below the target and concave above it, a portfolio isallocated in [25].14 Overall, using (7) as a utility measure may appear non-standard.

Indeed, the left-side derivative of hC(·, ·) at xT , i.e., the marginal utility, is zero, which might suggestthat the investor has no incentive to achieve xT . In contrast, in cumulative prospect theory à la [26],

12 Other constraints could be added, e.g., x(t) > 0.13 This measure (7) was also used in, among others, [16], [13] and [17]. A loss-averse utility function that is concave on

each side of the reference point (so, “similar" to (7)) was proposed in [22]. However, for the original parameters adoptedby [22], that function is only “lightly” concave and did not generate left-skewed distributions, see [15].

14 These authors solved the problem by splitting it into subproblems and found that the optimal strategy is one in whichthe investor takes on aggressive gambling strategies. The strategies computed in [23] and [24] still generate right skeweddistributions, which we deem not preferable by pension fund investors.

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the marginal utility at the target is positive. Despite this, (7) can be used for pension optimisation.Function hC(·, ·) is practically flat for yields x(T ) that are close to the left-hand side of xT and steepwhen the values of x(T ) are significantly smaller than xT . Thus, marginal utility is close to zero whenx(T ) approaches xT from the left and very large when x(T ) << xT . Hence, the investor using thisfunction will strictly avoid large losses and remain almost indifferent between pensions that are in the(left) vicinity of xT . This is the type of preferences that we ascribe to the pension investor for whom alow pension spells financial disaster. On the other hand, the convex shape of a prospect-theoretic utilityfunction means that it is steep when x(T ) is left-adjacent to xT and flattening for small x(T ). Thus theprospect-theoretic investor will be practically indifferent between medium and large losses. That is nota feature of our pension investor.

Figure 1. Risk-averse hM , prospect theoretic hP and the “double" concave hC performancemeasures for calibrated models.

In hC (see (7)), xT is a payoff-reference parameter rather than the yield target. It may understoodas a stochastic equivalent to a guaranteed minimum return from [27]. The behaviour of hC above xT istypical of risk-averse utility functions and mildly encourages investment above this value.

The above features of the function hC(·, ·) can be seen in Figure 1 where this utility measure isshown (solid line) along with the other commented on utility functions, for a plausible parameter set(see Appendix A). In this scale, Merton’s concave function looks like a straight line (see hM - bluedotted line). Clearly, the prospect-theoretic function hP (dashed line), which is convex to the left ofxT = $100, 000, attaches much smaller penalty to low pensions than the “double" concave hC .

4. Investment Strategies and Pension Distributions without Transaction Costs

We will first assume there are no transaction costs, b = 0. For this case, we will demonstratethe left-skewness of payoffs generated by a cautious-relaxed strategy and the corresponding controljerkiness needed to rebalance the portfolio. We will comment on these features in the context of theMerton investor’s payoff distributions and controls. We will also show that the prospect theoreticpayoff distributions are situated somewhere between Merton’s and cautious-relaxed. Because of that,

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they appear of lesser interest to pension savers (see [15]) and we will not delve into their sensitivity totransaction costs.

4.1. Optimal strategies

Let µ.(x, t), t ∈ [0, T ] denote a strategy for keeping an optimal proportion of shares. This strategymaps an observation x(t) at time t into the optimal number of shares u(t). (We claim that wealth x(t)

is more natural upon which to build a useable feedback strategy than state price density as in [22].) Toestablish the optimal strategy with b = 0, an investor will solve the following optimal control problem:

µ.(x, t) = arg max (9) subject to (5) and (3) . (10)

The Merton-optimal strategy for the proportion of shares, µM(x), which does not depend on t (seee.g., [18]), can be established through (10) with h(·) in (9) replaced by (8). A cautious-relaxed investorwill seek µM(x, t) through (10) with h(·) in (9) replaced by (7).

As said in the introduction, the Merton strategy µM(x) can be analytically established (seeAppendix C) while the cautious-relaxed µC(x, t) has been obtained numerically15 by the specialisedsoftware SOCSol from [12] (also, see [28]).

We will calculate the optimal strategy µC(x, t) for a parametrised pension management problem,described in Appendix A. Briefly, the initial outlay is x0 = $40, 000 and is to be managed for T = 10

years. If the volatility from the risky asset was eliminated and only the drift remained then $40,000would grow to $89,021.63 in 10 years (see (13)). Rounded up, this yields $100,000 which was selectedas the pension-reference level xT , not too easily obtained.

The strategies that solve a portfolio-investment problem of a cautious-relaxed investor with b = 0 areshown in Figure 2. The vertical axis is u(t), the proportion of wealth to be invested into shares and thehorizontal axis is the level of wealth x(t). The strategies are feedback (see [11]), they change with timeand state (wealth). It seems intuitive that an investor would adjust their allocations with performance.For example, if an investor’s fund is performing poorly, the investor would be well advised to shift theirinvestment towards more risky investments in order to at least have the chance of recouping losses. Thisis different to the application of a flat strategy obtained for the Merton investor, see (21) in Appendix C.

15 The problem with an analytical solution is that the as long as α in the utility measure (7) is just any number greater than 1,little can be said about a closed-form strategies and value functions. This is because t and x in V (t, x) (as in (18)) for theboundary problem with V (T, x(T )) = −(xT − x(T ))a, a > 1 appear non-separable. Nevertheless, even if V (x, t) wereobtained in an analytical form, a closed-form for the payoff-density function would still be an open problem. We alsonote that closed-form solutions in [22] were obtained for a similar but non-identical, utility function. More importantlytheir independent variable is not the current (observable) wealth but state price density.

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Figure 2. Control rules for xT = $100, 000 and t = 2, t = 7.

In more detail, we notice that the investment minimum, which occurs in this figure at u = 0, definestwo investment zones.(1) Fund values x(t) ∈ (0, xS(t)) where xS(t) is where u = 0. In this zone, the pension reference levelxT can only be reached by investing in the risky asset. Here, the investor must gamble to evade heavypenalties for falling short of xT .(2) Fund values x(t) > xS(t), from which xT can be reached by investing solely in the secure asset.In this zone, the investor strives to maximise their reward for exceeding xT , which is a less aggressiveprocess than in the first zone because κ ∈ (0, 1) in this zone (see (7)).

As said, the investor is relaxed about exceeding the reference level xT . If the fund value is such thatxT can be reached by investing all wealth in the secure asset, committing a fixed minimal portion ofx(T ) (e.g., $100) to the shares could, for large volatilities, result in a negative expected yield. Hence,if investment cannot be made smaller than a predetermined number of shares, then the zero investment(u(t) = µ(xS, t) = 0) may dominate this fixed minimal portion investment. For some details, seeAppendix D.

In Figure 3 a random selection of 20 (out of 100,000 used to generate Figure 4) time profiles of thefund values and the strategy realisations for t ∈ [0, T ] ≡ [0, 10] are shown for a Merton investor (“ME"- left panels) and a cautious-relaxed investor (“CM" - right panels). The strategy realisations uC(t) (inthe right bottom) are jerky (see [11]) as opposed to Merton’s which appear flat.16 This implies that acautious-relaxed investor will have to rebalance their portfolio very frequently, and expensively if b islarge. The shares’ exposure adjustments of the Merton investor are evidently much smaller.

16 The strategies are constant but some rebalancing is still needed to maintain a constant optimal proportional exposure ofthe risky asset, see [18] or [17].

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Figure 3. Fund value and strategy realisations.

4.2. Pension distributions

Figure 4 shows three pension payoff distributions, cumulative in the left panel and probability in theright panel. The distributions are obtained for a parameterised pension management problem, definedin Appendix A; in particular the reference payoff xT = $100, 000. The distributions for hM (the bluedotted lines) were derived analytically as a result of applying the optimal Merton investment strategy(see Appendix C). The solid lines correspond to the simulated payoffs obtained as outcomes of thecautious-relaxed strategy, see [15]. The dashed lines are payoff distributions for a strategy that maximisesthe prospect-theoretic function hP , shown in Figure 1. The last two distributions are obtained by usingthe respective strategy 100,000 times and measuring the obtained payoffs.

Figure 4. Fund value cumulative and probability distributions.

Both panels tell the same story because they refer to the same distributions. We show both becausesome distribution features such as quantiles are easier to demonstrate on the cdf graphs, while others suchas modes are easier to illustrate on the pdfs. The summary statistics for the Merton and cautious-relaxedpension distributions are presented in Table 1. (For convenience of comparisons with the pensions withtransaction costs, most of them are repeated in Table 2.)

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Table 1. Merton and cautious-relaxed pension yield distribution statistics.

Statitic Caut.-Relax. MertonMode of x(10) $93,790 $55,300Mean of x(10) $74,922 $86,596Median of x(10) $83,373 $73,082Std. dev. of x(10) $21,723 $55,042Coeff. of skew. –1.017 2.164P(x(10) > $105, 000) 0 0.273P(x(10) > $80, 000) 0.562 0.438P(x(10) < $40, 000) 0.1077 0.15

The hM (Merton’s) yield distribution is log-normal and right-skewed with the mode below $60,000.The probability of achieving a yield better than $105,000 is about 0.27. The median is $73,082 whichis less than the mean $86,596. The probability of attaining less than $40,000, which is the initial outlayin our parametrised problem, is about 0.15 and the probability of reaching a pension above $80,000 is0.438. The corresponding numbers for the hC yield distribution (cautious-relaxed) are as follows. Theprobability of achieving a yield better than $105,000 is 0; the median is $83,343, which is more thanthe mean $75,228. The probability of attaining less than $40,000 is about 0.11 and the probability ofreaching a pension above $80,000 is 0.562. The numerical measure of skewness17 indicates a largedifference between the Merton’s and cautious-relaxed distributions.

We do not list the numerical values for the prospect-theoretic distribution hP (dashed lines), which arein-between the Merton’s and cautious-relaxed. The fact that hP ’s mode dominates the one correspondingto hC is obvious; however, the probability of scoring a payoff greater than $80,000 is higher forthe cautious-relaxed pension, which – in our view – lessens the attractiveness of a prospect-theoreticdistribution for pension optimisers.

The comparisons of the probabilities of a disastrous yield below the initial outlay and those of a yieldabove $80,000 clearly point to hC as the utility measure that a pension optimiser will favour.

In the next section, the impact of a transaction cost on the skewness of the terminal fund valuedistribution will be examined.

5. Cautious-relaxed strategies and pension distributions with transaction costs

5.1. Optimal strategies

Let ζ(x, u) denote a strategy for an optimal share exposure’s corrections, or portfolio rebalancing.(So z(t), see (4), is a realisation of strategy ζ(x(t), u(t)) at time t.) The cautious-relaxed investor nowsolves this optimal control problem:

ζ(x, u) = arg max (9) subject to (4), (5) and (3) (11)

17 The skewness coefficient is calculated as E[(

X−meanstand. dev.

)3]. It provides a measure of asymmetry in the distribution.

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with b > 0 and h(·) as in (7).The computed cautious-relaxed strategies for b = 0.005 and b = 0.05 are displayed in Figures 5

and 6, respectively. Strategies and their outcomes for more values of b are commented on in Table 2.Figure 5 shows a sample of the strategies for a “small" transaction cost. The left panel shows them as

functions of wealth x(t) and the right panel – as functions of the proportion of wealth invested in sharesu(t). These strategies tell the investor which adjustment z(t) needs to be made to the current proportionof shares u(t), given wealth x(t). In other words, a value of z(t) says by which proportion of wealth thenumber of shares needs to be altered in a period (so, z(t) are rates); here, the period length is ∆ = 1/3.

Suppose the investor’s wealth in time 2 is x(2) = $40, 000 and no shares are owned so, u(2) = 0.Figure 5 recommends (see the dash-dotted line),

z(2) = ζ(x(2), 0) ·∆ = 2.1 · 1/3 = 0.7 .

This value is less than (but still quite close to) 0.79, which Figure 2 tells us is the correspondinginvestment into shares at t = 2 when there are no transaction fees.

Figure 5. Control rules for t = 2 and t = 7 in x(t) (left panel) and u(t) (right panel) forb = 0.005.

Now, suppose that the investor’s wealth at t = 7 is $85,000. Figure 2 advises u(x(7)) = 0.038 asoptimal when there are no transaction costs. Suppose that at this time 10% of the investor’s wealth is inshares. In Figure 5, we can read (see the dotted line) that the exposure needs to diminish by

z(7) = ζ(x(7), 0.1) ·∆ = −0.225 · 1/3 = −0.075 .

So, u(7) = 0.1 − 0.075 = 0.025. This is less than 0.038, the shares’ exposure when there are notransaction costs.

The above two numerical examples suggest that charging the shares’ updates b = 0.005 per perioddampens the investor’s behaviour.

Consider now a higher cost (b = 0.05). Figure 6 shows ζ(x(t), u(t)) projected into z(t), x(t) andz(t), u(t) in the left and right panels, respectively. The strategies are flatter than in Figure 5, which

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means that smaller adjustments are recommended. In particular, for x(2) = $40, 000 and if the originalnumber of shares is zero, Figure 6 recommends (see the dash-dotted line)

z(2) = ζ(x(2), 0) ·∆ = 1.575 · 1/3 = 0.525,

which is less than than 0.7, optimal when b = 0.005. As the value of the original number of shares waszero, z(2) = 0.525 is the number of shares to buy. This is obviously less than 0.79 when b = 0, readfrom Figure 2. Evidently, the investor restrains not only their share-adjustment size but also the riskyasset exposure.

Figure 6. Control rules for t = 2 and t = 7 in x(t) and u(t) for b = 0.05.

Figure 7. Sample of fund value, proportion of shares and shares’ adjustments for b = 0.01.

Figure 7 shows a sample of realisations of fund value, proportion of shares and shares’ adjustments foran even higher (than in Figure 6) transaction cost b = 0.01. A casual comparison of the first two panelsin this figure to Figure 3’s clearly documents that this transaction cost dampens the jerky behaviourof shares’ exposure corrections, (almost) without affecting the desirable yield’s clustering in the top ofits graph. It will be interesting to see if the left-skewness of the pension distribution for this cost willbe preserved.

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5.2. Pension Distributions

We will now see if the restrained share updating behaviour has an impact on the left-skewness of thepension yield.

The solid black cdf in Figure 8 (in the left panel) is the pension distribution for no transaction costs,which is the same distribution as shown in Figure 4. As documented by the dashed (red) and dash-dotted(blue) lines, for small and median transaction costs (b = 0.005 and b = 0.01), the left-skewness of thepension yield distributions is preserved. The dotted line is for b = 0.1 and looks almost symmetric, Thisclearly demonstrates that the left-skewness is lost for high costs.

Figure 8. Fund value cumulative and probability distribution functions for various valuesof b.

Table 2. Pension distributions statistics.

Statistic b=0 b=0.005 b=0.01 b=0.05 b=0.1 MertonMean of x(10) $74,922 $74,000 $73,657 $70,651 $68,493 $86,596Median of x(10) $83,373 $80,133 $78,815 $71,297 $67,966 $73,082Std. dev. of x(10) $21,723 $20,741 $20,464 $18,108 $15,640 $55,042Coeff. of skew. of x(10) –1.017 –0.8499 –0.78 –0.331 0.0158 2.164P(x(10) > $80, 000) 0.562 0.503 0.475 0.333 0.233 0.438P(x(10) < $40, 000) 0.1077 0.0938 0.0875 0.0561 0.0317 0.15

Examine Table 2, which provides aggregate statistics on the pension yield distributions for severaltransaction cost rates. The column marked ’b = 0’ gives the numbers for the costless caution-relaxedstrategy from Section 4. The last column is for the classic Merton strategy (21) without transaction costs.

As expected, the higher a transaction cost, the lower the mean and median of the pensionyield. Standard deviations also decrease as the costs grow, which indicates smaller variability ofthe yield for higher charges. Medians dominate means but their differences become small, whichis evidence of the distributions’ symmetrisation. The numerical measure of skewness also indicatesincreasing symmetrisation.

The last two rows of Table 2 provide measures of how well a strategy scores on the main two featuresof cautious-relaxed strategies: a high probability of yields proximate to the reference position xT and a

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(probabilistic) guarantee of low losses. We can see that charging b = 0.01 and less, delivers distributionswhere obtaining x(T ) ≥ $80, 000 is at least 0.475, which is more than using the Merton strategy (see lastcolumn). The low probability of losses (x(T ) ≤ $40, 000) is obtained for all cautious-relaxed strategies.However, higher charges than b = 0.01 make a cautious-relaxed strategy unattractive because of thedecreasing mean and a small probability of approaching the reference pension.

5.3. Advice

A question might be asked how high the transaction costs may be in “real-life". In the parametrisedproblem, the rates in equation (5) are expressed in annual terms. We have read in Figures 5 and 6 that thez-values for x(2) = $60, 000 and u(2) = 0 are 0.6 and 0.45 for b = 0.005 and b = 0.05, respectively. So,correspondingly, the trimester charges for these values of b would be 0.005 · 4/12 · 0.6 · $60, 000 = $60

and 0.05 · 4/12 · 0.45 · $60, 000 · 4 = $450.As a “rule of thumb”, our investor should use cautious-relaxed strategies if an annual cost aggregates

to less than $500, which would roughly correspond to b = 0.01.

6. Conclusion

This paper performs a further analysis of issues raised in [11] involving cautious-relaxed investmentstrategies obtained as optimal solutions to a stochastic optimal control problem of pension portfoliomanagement. It was shown there that a non-gambling investor can find left-skewed payoff distributionsfavourable that result from a utility measure kinked around a payoff reference point. However, theunderlying cautious-relaxed investment strategy realisations were jerky, which might have been due tothe absence of transaction costs. In this paper a sensitivity analysis of the cautious-relaxed strategies totransaction costs has been performed. It has been found out that realistic costs can mitigate the jerkinesswithout destroying the pension yield distribution’s left-skewness. For high transaction charges, using theclassic Merton strategy becomes attractive.

The above results allow us to propose that payoff distributions are important determinants ofadvantages between various investment strategies obtained as maximisers of some utility measures. Asmany distributions are analytically intractable the proposed approach, which is scalable to more complexmodels, should be of interest to funds’ managers.

Acknowledgments

I am indebted to Wilbur Townsend for insightful comments and editorial suggestions and to VladGaitsgory for a discussion on “fast" controls. I am also very grateful to three anonymous referees fortheir constructive comments and to the academic editor of this journal for final advice. All remainingerrors are mine.

Conflicts of Interest

The author declares no conflict of interest.

A. Parameters

As stated in the introduction, our solutions are numerical and, hence, parameter specific (yet scalableto more complex models). The hypothetical base-case scenario that we study involves an initial outlay

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x0 = 40, 000 invested in a pension fund to be collected as a lump sum after T = 10 years. Table 3 showsthe rest of the parameters that are utilised in this study.

Table 3. Selected Parameters

r α σ c

0.05 0.085 0.2 .005

With these parameters the fund will accumulate to

$40, 000e(r−c)T = $40, 000e(0.05−.005)10 = $62, 732.49 (12)

if all money is invested in the risk-free asset. This figure can be referred to as the guaranteed amount. Itshould be expected that a pension fund investor would require the pension yield to regularly surpass thisvalue.

However, if the volatility from the risky asset was eliminated and only the drift remained then $40,000would grow to

$40, 000e(α−c)T = $40, 000e(0.085−0.005)10 = $89, 021.63 (13)

in 10 years.

B. SOCSol and Yield Distributions

Due to the non-symmetric and non-smooth nature of the cautious-relaxed investor’s utility measure,an analytical solution to the investment problem is not available at this stage. We have resolved this bygenerating the optimal investment strategies using a program called SOCSol, developed in Matlab andintroduced in [29]. The program discretises optimal control problems and solves them using MarkovChains. See [12] and [13] for more details on SOCSol.

Different wealth-path realisations are generated within SOCSol using Monte Carlo simulation for agiven optimal investment strategy. The distributions of wealth at time T are obtained numerically byanalysing the generated realisations of x(T ) .

C. The Merton Investor

C.1. The Classic Utility Measure

An expected value of a concave utility function is the classic portfolio performance measure, see [4]and [30]. We will present the (known) optimal investment strategy and analyse the resulting payoffdistribution.

The choice of h(x(T )) in (6) is now the issue of concern. The classic Merton investor will use theconcave utility function (8) (compare [18]). Analytically, the optimal solution to their problem can easilybe found as a solution to the Hamilton-Jacobi-Bellman equation. First, we define a value function forthe portfolio problem:

V (t, x) = supu J(x(t), u) (14)

where J(·, ·) is expected utility as in (9). This gives the following Hamilton-Jacobi-Bellman equation:

max

[1

2u(t)2σ2x2Vxx(t, x) + (r + u(t)(α− r)− c)x(t)Vx(t, x) + Vt(t, x)

]= 0 (15)

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with boundary condition

V (T, x) =1

δxδ. (16)

Maximisation of (15) gives

u(t) = −α− rσ2

Vx(t, x)

xVxx(t, x)(17)

substituting into the Hamilton-Jacobi-Bellman equation (15) implies

0 = −1

2

(α− r)2

σ2

V 2x (t, x)

Vxx(t, x)+ (r − c)x(t)Vx(t, x) + Vt(t, x). (18)

The boundary condition set by (16) suggests the functional form for V (t, x):

V (t, x) = f(t)1

δxδ, f(T ) = 1 ; (19)

plugging this into (18) we are left with an ordinary differential equation

0 = δ

[1

2

(α− r)2

σ2(1− δ)+ (r − c)

]f(t) + f ′(t). (20)

This ODE is easily solved with a solution f(t) = eδ

((r−c)+ (α−r)2

2σ2 (1−δ)

)(T−t)

implying that (19) is the correctguess of the value function. Substituting this value function into (17) gives

u(t) =α− r

σ2(1− δ), (21)

which is the known “flat" optimal solution to the Merton investor’s problem (e.g., see [18]).The same proportion of wealth is always invested into the risky and risk-free asset. This seems to

be quite unintuitive in that you would expect an investor to change their strategy in order to adjust tothe performance of the portfolio. Some rebalancing of the portfolio will still be needed to maintain theconstant (21) but the corresponding transaction cost will be small.

C.2. The Distribution

While the cumulative and probability distributions for cautious-relaxed strategies are analyticallynon-tractable (and obtained numerically in this paper), the Merton payoff distribution shown as the(blue) dotted line in Figure 4 can be obtained in closed form.

We find18 that after fitting the optimal control (22) to the state equation (5), wealth x(t) is a GeometricBrownian Motion that follows

dx(t) = Mx(t)dt+ Σx(t)dw (22)

where M =(r − α)2

(1− δ)σ2+ r − c and Σ =

α− rσ (1− δ)

. This means that for an initial value x0, wealth x(t)

t ∈ (0, T ] is a log-normally distributed random variable with expected value

M(t) ≡ E(x(t)) = eMtx0

18 Compare the methods put forward in [31] and [5].

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and varianceΣ2(t) ≡ Var(x(t)) = e2Mtx2

0

(eΣ2t − 1

).

We also know that the wealth is the stochastic process

x(t) = x0 exp

((M − Σ2

2

)t+ Σw(t)

), t ∈ [0, T ] . (23)

Finally, the probability density function of wealth x(T ) is log-normal:

fXT (x; MT ,ΣT ) =1

xΣT

√2πT

exp

(−

(lnx− lnx0 − T (MT − 12Σ2T ))2

2Σ2TT

), x > 0 (24)

where MT , ΣT are M(t), Σ(t) evaluated at t = T .We know that a log-normal distribution is generically non-symmetric (which, in our view, causes a

problem of using strategy (21) by an investor seeking to avoid large losses) .Substituting the values given in Table 3 to the Merton solution given in (21) and using a δ = 0.05

gives an optimal u(t) of 0.921. This means that 92.1% of wealth at any time is invested in the riskyasset. Now that all the respective parameters have been set, we can evaluate the mean, variance andother statistics using the formulae given above, see Table 2.

The Merton investment strategy provides the distribution of payoffs shown as the (blue) dotted linein Figure 4. We see that the payoff is right skewed with relatively high probabilities of being below theoriginal investment and of being below the guaranteed amount, all analysed in Table 2.

D. Why Zero-Investment Can Be Profitable in This Model

It is important to note that some levels of wealth, which we denote xS(t), will virtually generateno investment in the risky asset. This will happen when the reference level payoff xT can be obtainedwithout the risky asset and if the risky-asset investment brings an expected value that is less than thesecure investment. For example if at time 6, x(6) = $83, 527, then u(τ) ≈ 0 for τ ∈ [6, T ]. Thisis because $83, 527e(0.05−0.005)(10−6) = $100, 000 so, xT is obtainable without the risky asset. For therelatively high volatility (σ = 0.2), the risky-asset investment of the size compatible with the adoptedgrid size (or constrained by a minimal investment), brings an expected value of marginal utility that isless than the secure gain and all funds19 will be allocated to the risk-free asset. We can say that wealthat time 6 has reached the secure-investment level xS(6). Using u(τ) = 0 for τ ≥ 6 from xS(6) causesevery x(τ) to the right of the x-intercept of the strategy graph in Figure 2, to be reached at the right timefor the investor to continue the strategy u(xS(τ)) = 0.

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